continuity of a function of two variables on a circle Let 
$f(x,y)=1$ on $x^2+y^2=1$ and 0 otherwise.
At what points $f$ is not continuous? The answer is, $f$ is discontinuous everywhere on the circle $x^2+y^2=1$.
Could anyone explain to me intuitively why? Rigorously how?
 A: That shows only that the function is not continuous at a single point on the circle.  More generally, a function, f(p) is "continuous" on a set if and only if it is continuous at every point on the set.  Further, a function, f(p), is continuous at p= a if and only if, we can make f(p) as close to f(a) as we please ($|f(p)- f(a)< \epsilon$ for any given $\epsilon$ by making p close enough to a ($|p- a|< \delta$).  Here, given any point, a, on the circle, so that f(a)= 1, there exist points arbitrarily close to a that are not on the circle and so have function value 0.  Another way of saying this is that, given any point a on the circle, any small circle, no matter how small, around a will contain points that are on the original circle and points that are not on it.
A: Editted Hint
Definition of continuity in 1D:

$f$ is continuous at $a$ iff $\ f(a_+) = f(a_-) = f(a)$

In this problem if $a$ is on the circle and $a_\epsilon$ is a point outside the circle with $|a-a_\epsilon|<\epsilon$, then we have
$$
f(a_\epsilon) = 0
$$
but
$$
f(a) = 1
$$
So not continuous.
